derivative of 1/x y

解

\frac{d}{d x} (\frac{1}{x} y)

- \frac{y}{x ^{2}} + \frac{\frac{d y}{d x}}{x}

解答ステップ

\frac{d}{d x} (\frac{1}{x} y)

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = \frac{1}{x}, g = y

= \frac{d}{d x} (\frac{1}{x}) y + \frac{d y}{d x} \frac{1}{x}

\frac{d}{d x} (\frac{1}{x}) = - \frac{1}{x ^{2}}

= (- \frac{1}{x ^{2}}) y + \frac{d y}{d x} \frac{1}{x}

簡素化

(- \frac{1}{x ^{2}}) y + \frac{d y}{d x} \frac{1}{x} : - \frac{y}{x ^{2}} + \frac{\frac{d y}{d x}}{x}

= - \frac{y}{x ^{2}} + \frac{\frac{d y}{d x}}{x}